OFFSET
0,5
COMMENTS
A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis.
A peak is an occurrence of the pattern UD.
A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
LINKS
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Algorithmica (2019).
FORMULA
G.f.: ( (-3*t^2+4*t+sqrt(-3*t^4+4*t^3+2*t^2-4*t+1)-1)/(3*t^2-4*t+1) + (2*t^3-5*t^2+4*t+sqrt(4*t^6-12*t^5+13*t^4-8*t^3+6*t^2-4*t+1)-1)/(-2*t^3+5*t^2-4*t+1) - (-5*t^2+4*t+sqrt(5*t^4-4*t^3+6*t^2-4*t+1)-1)/(5*t^2-4*t+1) - (-2*t^3-3*t^2+4*t+sqrt(4*t^6+4*t^5-11*t^4+8*t^3+2*t^2-4*t+1)-1)/(2*t^3+3*t^2-4*t+1) ) / (8*t).
a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2019
EXAMPLE
For n = 4 the a(4) = 5 paths are UHDU, UHDH, UUHD, HUHD, UHHD: in all these paths, 0 peaks, 1 hump.
For n=0..6 we have only paths with 0 peaks and 1 hump.
For n=7, we have a(7)=163. Among them, 160 paths with 0 peaks and 1 hump, and 3 walks with 2 peaks and 3 humps: UDUDUHD, UDUHDUD, UHDUDUD.
MAPLE
b:= proc(x, y, t, p, h) option remember; `if`(x=0, `if`(p+1=h, 1, 0),
`if`(y>0, b(x-1, y-1, 0, irem(p+`if`(t=1, 1, 0), 2), irem(h+
`if`(t=2, 1, 0), 2)), 0)+b(x-1, y, `if`(t>0, 2, 0), p, h)+
b(x-1, y+1, 1, p, h))
end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..35); # Alois P. Heinz, Jul 04 2019
MATHEMATICA
CoefficientList[Series[((-1 + 4*x - 3*x^2 + Sqrt[(-(-1 + x)^2)*(-1 + 2*x + 3*x^2)])/ (1 - 4*x + 3*x^2) - (-1 + 4*x - 5*x^2 + 2*x^3 + Sqrt[(-1 + x)^3*(-1 + x + 4*x^3)])/ ((-1 + x)^2*(-1 + 2*x)) + (1 - 4*x + 5*x^2 - Sqrt[1 - 4*x + 6*x^2 - 4*x^3 + 5*x^4])/(1 - 4*x + 5*x^2) + (1 - 4*x + 3*x^2 + 2*x^3 - Sqrt[1 - 4*x + 2*x^2 + 8*x^3 - 11*x^4 + 4*x^5 + 4*x^6])/(1 - 4*x + 3*x^2 + 2*x^3)) / (8*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 09 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jul 04 2019
STATUS
approved